Life is a process of becoming, a combination of states we have to go through. When people fail is that they wish to elect a state and remain in it. This is a kind of death.
Anaïs Nin, in D.H. Lawrence: An Unprofessional Study (and as found at the end of my PhD thesis)
I pulled over, and I thought about it. The wind was as cold as steel and sharp, invisible nails piercing through my hands, as I was filling the car up.
It was exactly then, as I was fiddling with the fuel dispenser at a filling station, that I thought about it once more.
We had been driving all over southern Wales, towards the setting sun, a race against the time, against a wild headwind, to see the sky blush orange and the sea swallow the remains of the day. The narrow road twisted up and down following the contour of the hills, a grey ribbon unfolding before us, a skin shed by an invisible snake, a trail for us to follow.
Directions, paths, cycles, and irreversible transformations. I thought about all of this as I sniffed the sweet organic smell of petrol. Harmful. Irresistible, like so many things in life. Like speed, and the fear and the thrill that go with it. What car racing is made of. Ayrton Senna once remarked that: “We are made of emotions, we are all looking for emotions, it’s only a question of finding the way to experience them. There are many different ways of experience them all. Perhaps one different thing, only that, one particular thing that Formula One can provide you, is that you know we are always expose to danger, danger of getting hurt, danger of dying“1.
Heat as a source of motion. Danger as the source of emotion.
I had enjoyed that long drive, allowing myself some fun with the last gearstick that my left hand would shift for quite a long time. A long straight dives down, then an uphill strech comes up as a sharp turn approaches: brake, change down and double-declutch -or, I’d better say as far as I’m concerned, do your best attempt at it-, steer, enter the corner, open the throttle again, and floor the pedal: it did not take much more than this to feel as close as ever to the world of car racing.
Something clicked and I stopped day-dreaming. The tank was full, petrol dripping from the nozzle of the fuel dispenser. Droplets drifted in the wind while falling down, and I felt for those lost hydrocarbons, lost and vaporised into the crisp air of an early spring day, somewhere in Wales.
And I thought about it once more.
I thought about entropy.
Illusions of stillness
Thermal engines, and their elegant profiles sketched on pressure-volume plots. It seems like yesterday, but it is a life ago. Secondary school, my brand-new driving licence in my wallet, and those long hours studying invisible gasses being compressed, expanding, at a frustratingly slow rate, for equilibrium to be attained.
Stillness. Only then does entropy remain unchanged.
My duel with thermodynamics continued at university. Fast-forward to those rainy days of November 2002 when the grey city was drenched, its underground was flooded, and its sewers were bursting at the seams. At first, I felt a mortal dread of that first-year physics course. The professor, a middle-aged stamp collector sporting a grey toothbrush moustache, relished the thought of inspiring terror in his students, and wielded his power in the most unlikely of ways, for examply by punishing students who mispronounced physicists’ names. You could easily fail an exam because of the tricky uy combination in the name Huygens.
Maybe that is why I ended up going to the Netherlands for my PhD. To learn to say “Huygens” the Dutch way. To visit the Provinces that could flourish during their Golden Age of tulips, trade and art. To see the canals criss-crossing Amsterdam. To meet the intense gaze of Vermeer’s Girl with a Pearl Earring, and taste the overripe fruits that defy time in Dutch still lives.
Quasistatic images, like those reversible processes.
No. I’m fooling myself. It’s nothing but an illusion. Even then, even when she was actually by my side, even on those tablecloths… entropy ruled supreme, as it always does. Loss, dissipation, and disorder. Look more closely: there’s always a fly on the cheese, or a bruise, a crack on the skin of the fruits. They’re about to rot.
Stillness does not befit a chemist, after all. Chemical equilibria, one of the defining features of our science, hide microscopic, frantically reversible transformations occurring at a blistering pace, all under the cover of a macroscopic invariance.
So, a chemist’s inner balance is dynamically stationary, reminiscent of what Tolstoj writes in The Death of Ivan Ilych: “He in his madness prays for storms and dreams that storms will bring him peace“.
The howling gale was sweeping the Welsh coast.
I drove off. I turned the ignition on.
Time to go, follow the fuel and its flow.
Dissipation and multiplicity
I have always regarded the Second Law of thermodynamics with a mixture of awe and distrust. I am aware that this law stands for something powerful and ubiquitous, but I have always believed that it is, in some respects, utterly incomprehensible at the same time. Part of this gut feeling probably boils down to the countless ways of expressing, defining, interpreting the law. If you want to discover more about the confusingly multifarious nature of the Second Law, the Web will provide loads of notes, course handouts, etc. : just have a look out there, for example on this page.
What most angered me was the concept of efficiency. Don’t laugh at me! For reasons that I struggle to explain, I have always felt for thermal engines, toiling and sweating and ticking over, only to convert into work just a fraction of the energy extracted from, say, petrol. Dissipation was inevitable, like a thermodynamic curse placed on engines, a fact of nature which, I firmly believed, was deeply unjust.
So, despite passing all my physical chemistry and physics exams with flying colours, I always felt profoundly uncomfortable with entropy and the Second Law until I attended the fourth-year course on statistical thermodynamics. Boltzmann’s formula changed my outlook. It reads like this:
S = kB ln W
where kB is Boltzmann’s constant and W stands for the number of microscopic states consistent with a given macroscopic state. How to understand this concept? Suppose you want to describe an ant colony: you can choose to approach this task at a macroscopic level (how big the mound is, its temperature…) or try and describe the colony on the basis of the position and the speed of each ant. The overall appearance of the ant colony will not change for countless equivalent sets of positions and speed of all the individual ants. Well, these equivalent sets are by far and large a good example of what W means.
(At any rate, note that here we encounter once more the micro/macro duality that pervades all chemistry).
Another autumn, another clash with thermodynamics, another tryst with the Girl with a Pearl Earring. It was the year 2005, and I was preparing for my first adventure abroad, the Erasmus exchange project at Leiden University. I remember slipping handouts on the conjugation of Dutch verbs into the pages of the reference textbook of that course, the venerable Fundamentals of Statistical and Thermal Physics, by Frederick Reif.
I remember defying entropy with her. Perhaps.
One day, the lecturer stopped halfway a sentence, stared at us and said: “This formula is carved on Boltzmann’s tombstone“.He paused for a while, as if had forgotten what he was to say. Then, he quipped, grinning proudly : “Physical chemists never die, they tend to the maximum entropy”.
Equilibrium as the maximum number of equivalent states. Electing one is a kind of death.
Ode to entropy
A few days ago, on a sleepless night, I found myself thinking about Boltzmann’s formula and those long-lost days at university.
I closed my eyes. My mind strayed as I was humming a tune…
…infinité de destins
on en pose un
qu’est-ce qu’on en retient?…
…an infinity of destinies
we set one aside
what remainder will we keep?
…entropy, entropy everywhere once more, entropy blowing sand across the desert and I was wondering what we keep, what we know, when we choose, take a turn, leave a path. I looked at my hands. I saw words as if tattoed on my skin.
occorrono troppe vite per farne una
“Too many lives are needed to make just one…”2
I startled. It was just an ink stain, my fountain pen had smeared my fingers, again.
Not a wink of sleep. My mind drifted away to the Aegean Sea, and somehow these forgotten words came ashore, like a message in a bottle, like a sudden flash:
μή, φίλα ψυχά, βίον ἀθάνατον
σπεῦδε, τὰν δ᾿ ἔμπρακτον ἄντλει μαχανάν. 3
Among others, here is my favourite translation:
“O my soul, do not aspire to infinite life, but exhaust the limits of the possible”.
The limits of the possible…like in racing, the limits of grip define how fast you can drive, how dangerously you can live. Listen to Ayrton Senna once more: “you think you have a limit. And you then go for this limit and you touch this limit, and you think, ‘Okay, this is the limit.’ As soon as you touch this limit, something happens and you suddenly can go a little bit further.” 1
Treading the fine line between grip and spin seems the price to pay to live life to the full. Like a ± sign, like the uncertainty of measurement, the secret lies in that blurry space, a shimmering haze surrounding our lives and defying all attempts to define them.
I thought about Boltzmann’s formula once more, and its being a special case of a more generic formulation of statistical entropy known as Shannon entropy related to information theory.
The higher the entropy of a state, the higher its probability, but also its microscopic randomness, which means that we know less about it. Yet, see it the other way: it is uncertainty that unlocks multiple possibilities, multiple equivalent states.
Not convinced? Take Boltzmann’s formula. S = 0 when W = 1. Indetermination disappears when there is no multiplicity, and then we know everything. Or maybe not, because we can reach S = 0 only without being alive to experience it. To put it bluntly, death is the only case when there is just one possible state. What a price to pay.
Well…”Electing one is a kind of death” once more, right?
So, Boltzmann’s formula can really come into its own in our everyday life, as this is not a dry mathematical expression, but something within grasp. For example, think about this interpretation: being alive, having in other words S ≠ 0, implies that there is more than one equivalent microstate. Well, I’d like to imagine that these microstates are the countless permutations and combinations of mood, thoughts, ideas that each of us experiences at a given moment in life.
In other words, we should stop despising entropy. Look it in the face: after all, entropy was coined from the Ancient Greek ἐντροπία, meaning “a turning towards”.
Yes, I hear you shrug and say: “Brilliant metaphors, a fine piece of writing, but in the end that’s just empty talk”. Fair enough. Yes, entropy may well be a spell cast on us, and yet everything changes when we make most of it, when we acknowledge and harness it. Now, the key question is, how to attain an equilibrium embodying the maximum multiplicity of accessible lives at a given moment? At the same time, how to make sense of and shoulder the ever heavier burden of the alternative routes that we have not followed, of all the paths left behind at the many crossroads, along the arrow of time?
We need to talk about chemistry
In my realms of chemical fantasies, I imagine that the arrow of time can put on peculiar disguise, such as the extent of reaction or the reaction coordinate, turning into transforming matter and space, respectively.
The extent of reaction is, as defined by the IUPAC Gold Book, an “extensive quantity describing the progress of a chemical reaction equal to the number of chemical transformations, as indicated by the reaction equation on a molecular scale, divided by the Avogadro constant (it is essentially the amount of chemical transformations)”.
So, if you take it word by word, the extent of reaction, indicated by the graceful letter ξ, is nothing but the amount of stuff being transformed, corrected by the stoichiometric coefficient (“as indicated by the reaction equation on a molecular scale”), and expressed in moles. Why am I talking about ξ ? Simply because it will unlock yet another reincarnation of the infamous Second Law, which sneaks in and determines the conditions for chemical equilibrium, and the direction of chemical reactions.
To understand this, we need to face the so-called Gibbs free energy, G, something that is dearly beloved by chemists. A lightning sketch of what it means? Take a battery, any battery, and read its voltage. Guess what? You are looking at a masked form of G.
For those who prefer the nuts and bolts, G packs up in a neat way all sorts of variables that can come into play in chemical transformations. Its definiton is:
G = H – TS
where H is the enthalpy, T the (absolute) temperature, and S our dear entropy. So far, so good, all like in a textbook. What I would like to point out is this:
- entropy always matters, unless the temperature is zero.
- entropy is not the only thing that matters, when talking about free energy…
- …but entropy is to be inteded as entropy of the chemical system, so an “internal” state function of the system, while enthalpy is transferrable, energy exchanged with the surroundings.
Chemical reactions and their inner entropy, along with another form of energy that impacts what surrounds us. The randomness within together with the energy that we can share with those who are closest to us.
Chemistry can be so close to the human scale.
Unveiling what G really depends on is a good way of understanding free energy: pressure, temperature, and number of molecules are its natural variables. Let us focus on the latter, because it is, arguably, the ‘most chemical’ of the three. If we forget about the first two variables, keeping them constant (or making the assumption that they shall be), we start to understand why the extent of reaction ξ comes into play. We can imagine that ξ is like a dial letting us play with the concentration of reactants and reagents of a certain chemical reaction, changing their amount, which means moving the reaction forwards and backwards, and so time, at least in our thought experiment. G will respond accordingly: think about the focussing knob of binoculars, there will be a position giving the sharpest focus, while turning left and right will both give an unfocussed image.
Focussing means turning the knob until we reach the minimum blur.
Equilibrium means tuning ξ until we reach the minimum G.
This change in G as a function of ξ (the partial derivative of G with respect to ξ) is the free energy of reaction (watch out for the subscript r) which is accounted for by comparing free energies of products and reactants. This equals zero at equilibrium.
Let’s sum it up with a bit of maths and graphs:
Look at that last equation: ΔrH = TΔrS. Chemical equilibrium, the condition where the inner and outward energy associated with the chemical reaction are evenly balanced, without necessarily being equal to zero.
A dynamic levelling out of our randomness within, and the warmth that we give -or take.
Moreover, do not forget that we have set the composition of reactants and products by choosing the value of the extent of reaction minimising G. Is the chemical equilibrium then a boring stasis? Not really: two-way chemical transformations continue frantically and you can imagine that it is this ease of mutual interconversion of products into reactants and viceversa that embodies the “maximum randomness” corresponding to the minimum G. If we start from the reactant A, we can claim a much more detailed knowledge of what is in our flask than at equilibrium, when the molecules of A and B keep changing identity, despite being in macroscopically fixed proportions (set by the extent of reaction, which is the equilibrium constant in disguise).
Could chemistry be the most unlikely signpost of a dynamic peace, a waymark to happiness?
Shedding the old colours
Chemical equilibria are achieved through chemical transformations, and there is no better time of the year to talk about this topic than spring, the season of changes, of renewal. If life had a birthday, that would fall sometime in April. Yes, Eliot’s infamous “cruellest month”4, which turns wonderful when the “increase of the density of lived time may be found in those days of alternating sun and rain, […], when plants grow, almost visibly, several millimetres or centimetres a day. These hours of spectacular growth and accumulation are incommensurate with the winter hours when the seed lies inert in the earth” 5. The time of the year when birds migrate north, and wear their brightest plumage. They moult.
So does this blog: it sheds its worn winter feathers, sporting this new theme, and a new header image. Let me stop to acknowledge the work of a dear friend, fellow blogger, who took up the daunting challenge of turning my fuzzy ideas into a picture. Thanks ever so much, M.
This change accompanies yet another metamorphosis, another transition, a sudden gust of wind that makes my own flickering flame tremble and shake.
Like the atoms in a transition state…
…it was as I was listening to a song by the Italian band Baustelle that I thought about thermodynamics once more, on a sleepless night. Its title is La natura, (“Nature”), and the lyrics say it all:
L’unico modo per mostrare a tutti la felicità.
E’ la metamorfosi, la sola possibilità.
Ne sono sicura, muove la natura e la biologia
There’s just a single way of letting them all see
Only a metamorphosis will show how happy you can be
That’s a fact of nature of biology I can’t be wrong.
At a first glance, chemical reactions seem to imply that it is the outcome of the metamorphosis that matters the most. Beginning and end, products and reactants: what we had before and what we have in our flasks now; what we used to be and what we will be.
However, as my road trip in Wales taught me, with the many lengthy detours we had to take, there are countless paths between two given points.
Charting the course
Here we finally encounter the other form of (as I see it) chemical arrow of time, the reaction coordinate, which, by far and large, has a geometric meaning, along with my suggestion to interpret it as disguised time. The reaction coordinate represents the change in a chosen geometric feature of a reacting molecule (say, the distance between two atoms, or the angle between them) which can be a proxy for the progress of the reaction. In a sense, the extent of reaction was a reaction coordinate of sorts, referring to reacting stuff, and not geometry. Also, we note the macro/micro distinction again: the reaction coordinate is much more on a microscopic scale than the extent of reaction, which is rather an accounting tool to keep track of the amount of transforming matter.
Now, our discussion of G taught us that free energy plays a key role in chemistry, and all the more so when talking about reaction pathways, the trajectories of transformations. Say that the hydrogen molecule H-H is falling apart: we can imagine that H on the left and H on the right start to move farther and farther until they become loose. It takes energy to do so, and this “effort”, expressed in the form of free energy, can be plotted as a function of the distance between H and H.
At the top you will encounter the transition state.
Reaction energy diagrams are not restricted to this classical xy plane of textbook plots: reaction coordinates can be more than one, giving rise to geometric hypersurfaces. Take the molecule H-H again, and imagine mishandling it in all possible ways, by pulling, twisting, bending the poor thing as you try to break it apart.The transition state is by definition at the top of saddle points. This precise localisation also comes in with a well-defined configuration and a 50-50 of forming the reactants or the products of a reaction.
Too neat, too clear. I had rather talk about activated complexes instead. It takes energy to get there, to reach those fleeting arrangements hovering close to the highest point of the trajectory which transforms reactants into products. Life is on the edge up there, but what a scenery one can admire! On top of that, one must have a head for heights to be an activated complex, teetering on the brink of a headlong fall backwards, or poised to plunge forward at a breakneck speed. You must be a hybrid of past and present, an entity without clear identity. I quote this beautiful, almost lyrical description from Wikipedia: “in other words, [the activated complex] refers to a collection of intermediate structures in a chemical reaction that persist while bonds are breaking and new bonds are forming. It therefore represents not one defined state, but rather a range of transient configurations that a collection of atoms passes through in between clearly defined products and reactants.”
Steer your reaction course, your racing line, the direction followed by the metamorphosis, climb the uphill stretch to the saddle point, become an activated complex, and then stop at the top of the pass, and enjoy the commanding view onto your energy landscape. Feel the wistful nostalgia for the reactant state left behind, be tempted by the descent backwards, yield to it, or dive down and cross the barrier into an unchartered territory,into the unexplored coordinates on the contour map…
La trajectoire de la course
Et ton message à la Grande Ourse
Un instantané de velours
Même s’il ne sert à rien
The trajectory of the race
and your message to the space
the sweetest picture we can take
though meaningless at all.
The path to the products may well follow a single course, and yet all other trajectories will combine, and plot your graph.
And entropy will account for all the lines that part.
Footnotes & Acknowledgements
V.B. is gratefully acknowledged for the featured image, and B.P. for the photograph closing the post.
- Eugenio Montale, L’estate (“Summer”).
- Pindar, Pythian 3, lines 61-62
- From the first lines of The Waste Land:
April is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain
- John Berger, Keeping a Rendezvous